In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" quantum electrodynamics.
The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via the Abelian Higgs mechanism.
The model consists of a complex scalar field
minimally coupled to a gauge field
This article discusses the theory on flat spacetime
(Minkowski space) so these fields can be treated (naïvely) as functions
The theory can also be defined for curved spacetime but these definitions must be replaced with a more subtle one.
The gauge field is also known as a principal connection, specifically a principal
The dynamics is given by the Lagrangian density
where This model is invariant under gauge transformations parameterized by
is an infinitesimal change of trivialization, which generates the finite change of trivialization
In physics, it is customary to work under an implicit choice of trivialization, hence a gauge transformation really can be viewed as a change of trivialization.
If the potential is such that its minimum occurs at non-zero value of
, this model exhibits the Higgs mechanism.
This can be seen by studying fluctuations about the lowest energy configuration: one sees that the gauge field behaves as a massive field with its mass proportional to
times the minimum value of
As shown in 1973 by Nielsen and Olesen, this model, in
dimensions, admits time-independent finite energy configurations corresponding to vortices carrying magnetic flux.
The magnetic flux carried by these vortices are quantized (in units of
) and appears as a topological charge associated with the topological current
μ ν ρ
These vortices are similar to the vortices appearing in type-II superconductors.
This analogy was used by Nielsen and Olesen in obtaining their solutions.
A simple choice of potential for demonstrating the Higgs mechanism is The potential is minimized at
This produces a circle of minima, with values
gauge symmetry containing a scalar field
coupled to a gauge field
is valued in a representation space of the gauge group
, making it a vector; the label of "scalar" field refers only to the transformation of
under the action of the Lorentz group, so it is still referred to as a scalar field, in the sense of a Lorentz scalar.